黎曼流形与图上的协方差与主成分分析
Covariance and Principal Component Analysis on Riemannian Manifolds and Graphs
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中文总结 AI 辅助
本文提出黎曼流形与加权图上的协方差及主成分分析方法，通过热核向量场映射实现无穷主成分捕捉非线性特征，并给出RiePCA与GraphPCA两种计算实现。
中文摘要 AI 辅助
我们为有界几何的黎曼流形上的概率测度发展了协方差和主成分分析（PCA）的概念，并为加权简单图顶点集上的分布建立了离散对应版本。我们不是将变异性锚定在弗雷歇均值上（在该背景下其唯一性和有效性可能失效），而是考虑关于每个点的变异性。这导致流形上的每个点通过热核导出的向量场来表示，从而将底层流形映射到向量场的希尔伯特空间，在该空间中协方差和PCA从欧几里得设置中继承而来。该公式的一个显著特征是通常能获得无穷多个主成分，能够捕捉高度非线性的几何特征和变异模式。我们证明了该映射到向量场空间的嵌入定理，并发展了该理论的两种计算简化：RiePCA，一种基于有限多个点上支撑的向量场的有限维简化；以及GraphPCA，一种针对加权图上测度的离散公式，其中向量场为边分配方向和大小。数值示例和实验展示了所提方法在合成数据和真实数据上的行为。
英文摘要
We develop notions of covariance and principal component analysis (PCA) for probability measures on Riemannian manifolds of bounded geometry and a discrete counterpart for distributions on the vertex sets of weighted simple graphs. Rather than anchoring variation at the Fréchet mean, whose uniqueness and usefulness can fail in this context, we consider variation about every point. This leads to a representation of each point on the manifold by a vector field derived from the heat kernel and thus to a map of the underlying manifold into a Hilbert space of vector fields, in which covariance and PCA carry over from the Euclidean setting. A distinguishing feature of this formulation is that one typically obtains infinitely many principal components, capable of capturing highly nonlinear geometric features and patterns of variation. We prove an embedding theorem for the mapping into the space of vector fields and develop two computational reductions of the theory: RiePCA, a finite-dimensional reduction based on vector fields supported on finitely many points, and GraphPCA, a discrete formulation for measures on weighted graphs in which vector fields assign orientations and magnitudes to edges. Numerical examples and experiments illustrate the behavior of the proposed methods on both synthetic and real data.
发表机构
- Universidade Federal de São Carlos（圣卡洛斯联邦大学）
- Florida State University（佛罗里达州立大学）
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